8.1.
Riesz Representation Theorem 1
Chapter 8. The Lp Spaces:
Duality and Weak Convergence
Section 8.1. The Riesz Representation
for the Dual of Lp , 1 ≤ p < ∞
Note. In this section, we introduce the idea of a linear functional on a linear
space and find that the set of all bounded linear functionals on a given linear space
is itself a linear space (called the dual space of the original space). The Riesz
Representation Theorem classifies bounded linear functionals on Lp (E) and allows
1 1
us to show that the dual space of Lp(E) is Lq (E) where + = 1 and 1 ≤ p < ∞
p q
(recall that such p and q are called conjugates).
Definition. A linear functional on a linear space X is a real-valued function T on
X such that for g and h in X and α and β real numbers, we have T (αg + βh) =
αT (g) + βT (h).
Note. For E measurable, 1 ≤ p < ∞, q the conjugate of p, and for g ∈ Lq (E),
R
define T on Lp (E) by T (f ) = E gf for all f ∈ Lp (E). By Hölder’s Inequality,
gf is integrable and so T is defined. Since integration is linear, then T is a linear
functional. Also by Hölder’s Inequality, for all f ∈ Lp (E), |T (f )| ≤ kgkq kf kp . This
will imply that T is a “bounded linear functional” on Lp . The Riesz Representation
Theorem states that every bounded linear functional on Lp is of the form of T .
8.1. Riesz Representation Theorem 2
Definition. For a normed linear space X, a linear functional T on X is said to be
bounded if there is M ≥ 0 for which |T (f )| ≤ M kf k for all f ∈ X. The infimum of
all such M is called the norm of T , denoted kT k∗ :
kT k∗ = inf {M | |T (f )| ≤ M kf k}.
f∈X
R
Note. The linear functional T defined above as T (f ) = E
gf is bounded by kgkq .
Note. For T a bounded linear functional on X, for all f, h ∈ X we have |T (f ) −
T (h)| ≤ kT k∗ kf − hk. So if {fn } → f with respect to the norm k · k, then
{T (fn )} → T (f ) in R.
Note. In Exercise 8.1, it is to be shown that
kT k∗ = sup{|T (f )| | f ∈ X, kf k = 1}.
If kf k < 1, then kf /kf kk = 1 and |T (f )| = |T (kf kf /kf k)| = kf k|T (f /kf k)| ≤
kf kkT k∗ < kT k∗ . So we can also say
kT k∗ = sup{|T (f )| | f ∈ X, kf k ≤ 1}.
Proposition 8.1. Let X be a normed linear space. Then the collection of bounded
linear functional on X is a linear space with k · k∗ as a norm. The normed linear
space of bounded functionals is called the dual space of X, denoted X ∗.
Proof. Problem 8.2.
8.1. Riesz Representation Theorem 3
Proposition 8.2. Let E be measurable, 1 ≤ p < ∞, q the conjugate of p, and
R
g belong to Lq (E). Define the functional T on Lp (E) by T (f ) = E gf for all
f ∈ Lp (E). Then T is a bounded linear functional on Lp (E) and kT k∗ = kgkq .
Note. We will see that the converse of Proposition 8.2 also holds. That is, every
bounded linear functional of Lp is of the form of T .
Proposition 8.3. Let T and S be bounded linear functionals on a normed linear
space X. If T = S on a dense subset X0 of X, then T = S on X.
Lemma 8.4. Let E be measurable and 1 ≤ p < ∞. Suppose g is integrable
R
over E and there is M > 0 such that | E gf | ≤ M kf kp for every simple function
f ∈ Lp (E). Then g ∈ Lq (E) where q is the conjugate of p. Moreover, kgkq ≤ M .
Theorem 8.5. Let 1 ≤ p < ∞. Suppose T is a bounded linear functional on
Lp ([a, b]). Then there is a function g ∈ Lq ([a, b]), where q is the conjugate of p, for
R
which T (f ) = [a,b] gf for all f ∈ Lp ([a, b]).
Note. The Riesz Representation Theorem extends Theorem 8.5 from [a, b] to
general measurable set E.
8.1. Riesz Representation Theorem 4
Riesz Representation Theorem.
Let E be measurable, 1 ≤ p < ∞, and q the conjugate of p. Then for each
R
g ∈ Lq (E), define the bounded linear functional Rg on Lp(E) by Rg (f ) = E gf
for all f ∈ Lp (E). Then for each bounded linear functional T on Lp (E), there is a
unique function g ∈ Lq (E) for which Rg = T and kT k∗ = kgkp .
Note. Proposition 8.1 and the Riesz Representation Theorem combine to show
1 1
that the dual space of Lp (E) is Lq (E), where + = 1 for 1 ≤ p < ∞. Surprisingly,
p q
the dual space of L (E) is not (in general, at least) L1(E). That is, there is a
∞
bounded linear functional on L∞(E) (for the case E = [a, b]) that is not of the
R
form T (f ) = [a,b] gf where g ∈ L1(E). The dual space of L∞(E) is given in the
Kantorovich Representation Theorem (Theorem 19.7) in the general setting of a
measure space.
Note. In the event that p = q = 2, we see that the space L2(E) is “self dual.”
The space L2(E) is special in other ways—it is the only Lp space on which an inner
product can be defined and is an example of a Hilbert space.
Revised: 3/3/2017
